Lecture 6

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					Lecture 6                          Digital Electronics – Karnaugh Maps                                              A.D


Simplifying Boolean Expressions

Simplify using Boolean algebra: AB  AB .

AB  AB        Original expression
A( B  B )    After applying distributive law
A1            After applying inverse law
A             After applying identity law
Therefore, AB  AB  A .

Karnaugh Maps (K –Map)

To use a Karnaugh map to simplify an expression:
   1. Draw a “map” with one box for each possible product term in the expression. The boxes
       must be arranged so that a one-box movement either horizontal or vertical changes one
       and only one variable. See Figure 1.
   2. For each product term in the expression to be simplified, place a checkmark in the box
       whose labels are the product's variables and their complements.
   3. Draw loops around adjacent pairs of checkmarks. Blocks are "adjacent" horizontally and
       vertically only, not diagonally. A block may be "in" more than one loop, and a single
       loop may span multiple rows, multiple columns, or both, so long as the number of
       checkmarks enclosed is a power of two.
   4. For each loop, write an unduplicated list of the terms which appear; i.e. no matter how
       many times A appears, write down only one A.
   5. If a term and its complement both appear in the list, e.g. both A and A , delete both from
       the list.
   6. For each list, write the Boolean product of the remaining terms.
   7. Write the Boolean sum of the products from Step 5; this is the simplified expression.

Karnaugh Maps for Expressions of Two Variables
Example 2:
Simplify the above expression ( AB  AB ) using K – Map.

Solution:
    Draw a rectangle and divide it so that there is a row or column for each variable and its
       complement.
    Place checks in the boxes that represent each of the product terms
       of the expression.
    Draw a loop around adjacent pairs of checks.


The loop contains A, B, A, and B . We remove one A so that the list is
unduplicated. The B and B "cancel," leaving only A, which is the
expected result: AB  AB  A .

Example 3: simplify AB  AB  AB
  Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation –
                                                  G6601/S-I/Y-09
Lecture 6                          Digital Electronics – Karnaugh Maps                                              A.D


Solution
            Place a check in the A B area.
            Place a check in the AB area.
            Place a check in the A B area.
            Draw loops around pairs of adjacent checks.


Because there are two loops, there will be two terms in the simplified
expression. The vertical loop contains A , B, A , and B . We remove one A to make an
unduplicated list. The B and B cancel, leaving the remaining A . From the horizontal loop we
remove the duplicate B , then remove A and A leaving only B in the second term. We write the
Boolean sum of these, and the result is A  B , so: AB  AB  AB  A  B

Expressions of Three Variables

Recall that an essential characteristic of a Karnaugh map is that moving one position horizontally
or vertically changes one and only one variable to its complement. For expression of three
variables, the basic Karnaugh diagram is shown below.

As with the diagram for two variables, adjacent squares differ
by precisely one literal. The left and right edges are
considered to be adjacent, as though the map were wrapped
around to form a cylinder.

Example 4 shows deriving a circuit from the truth table using the sum of products method,
simplifying the sum of products expression, and drawing the new, simpler circuit.




Figure 4. a) A truth table with product terms, b) the resulting sum-of-products expression (min terms), and c) the
equivalent digital logic circuit.

The Karnaugh map for the expression in Figure 4 b is shown below. In this Karnaugh map, the
large loop surrounds ABC and ABC ; note that it "wraps around" from the left edge of the map to
the right edge. The A and A cancel, so these two terms simplify to BC.
  Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation –
                                                  G6601/S-I/Y-09
Lecture 6                          Digital Electronics – Karnaugh Maps                                              A.D




 A BC is in a cell all by itself, and so contributes all three
of its terms to the final expression. The simplified
expression is BC  ABC . Simplified circuit is shown in
Figure 6.



                                                       Figure 6. Simplified circuit for the truth table of Figure 4a.

Example 5: Generate a sum of product expression from the truth table in Figure 7a, simplify it
using K - Map and draw a simpler circuit.




Figure 7: a) A truth table with product terms, b) the resulting sum-of-products expression, and c) the equivalent
digital logic circuit.

Figure 8 shows the K-Map of the expression.




                           Figure 8: Karnaugh map for the expression of figure 7




  Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation –
                                                  G6601/S-I/Y-09
Lecture 6                          Digital Electronics – Karnaugh Maps                                              A.D


After removing duplicates, the large loop contains A and A and also C and C ; these cancel. All
that's left after removing the two complement pairs is B. The small loop contains B and B , which
are removed, so it yields AC. The simplified expression in Figure 7b is
                                       B  AC



Figure 9: Simplified circuit equivalent to Figure 7c.




Electronic Logic
Electronic logic circuits work with two levels of voltage:
    Low: 0 V
    High: The positive supply voltage. In most logic circuit, „high‟ is always 5 V.
Usually the low voltage level corresponds to logical „0‟ and the high level to logical „1‟.

Logic Circuits have practical applications. A simple example is shown below.
Example 6

                                                         There are two switches in the circuit below. There
                                                         is one lamp. The circuit has two binary inputs and
                                                         one binary output.

                                                         There are four possible ways in which the two
                                                         switches can be set:
                                                         1. A open and B open: lamp off.
                                                         2. A closed but B open: lamp off.
                                                         3. A open but B closed: lamp off.
                                                         4. A closed AND B closed: lamp ON.

                                                The action of
                                                the circuit can
                                                be summarized
by representing the binary states of inputs and output by „0‟
and „1‟. For the switches, 0 = „switch open‟ and 1 = „switch
closed‟. For the lamp, 0 = „lamp off‟ and 1 = „lamp on‟.

The four states of the switches can be set in a truth table.



Example 7: Sprinkler Control
A system is designed to turn on a garden water sprinkler when the soil is dry, but not when the
sun is shining. A soil moisture sensor A has outputs 0 = moist, and 1 = dry. A light sensor B has

  Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation –
                                                  G6601/S-I/Y-09
Lecture 6                          Digital Electronics – Karnaugh Maps                                              A.D


outputs 0 = dull, and 1 = sunny. For the sprinkler S, 0 = off, and 1 = on. The truth table for the
system is shown below. Draw the logic circuit for the control.
                                                   Inputs                  Outputs
                                            A                  B               S
                                            0                  0               0
                                            0                  1               0
                                            1                  0               1
                                            1                  1               0
Solution:
The third line down corresponds to „dry soil and dull day‟, the ideal conditions for watering. For
S = 1, we see that A = 1 and B = 0. As a logic equation, this is written: S  AB




Note:
Building logic circuits is simple. All the logic gates and other more complicated logic circuits
that you might need are available as integrated circuits (ICs). There are two commonly used
„families‟ of logic ICs:
     TTL - short for transistor-transistor logic. This runs on 5 V, so it needs a regulated
        power supply. TTL type numbers all begin with „74‟ so this is sometimes known as the
        74XX series. There are various types of TTL, of which the Low Power Schottky type has
        almost replaced the original 74XX series. 74LSXX ICs need less power than the 74XX
        type.
     CMOS - complementary Metal Oxide Semiconductor. These types have numbers
        ranging upward from 4000, so are sometimes known as the „4000‟ series. There is also
        the „4500‟ series. Members of both series run on any voltage between 3 V and 15 V.
     CMOS is slower than TTL but requires less current. It has the additional advantage that it
        does not require a regulated power supply.
Both TTL and CMOS are packaged as double in- line ICs. They usually have 14 or 16 pins.
There are four AND gates to the 7408, 74LS08, and 4081 ICs. The four gates share the power
supply pins. TTL inputs may be left unconnected. An unconnected input behaves as if it has a
high (=‟1‟) signal applied to it.




  Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation –
                                                  G6601/S-I/Y-09
Lecture 6                          Digital Electronics – Karnaugh Maps                                              A.D


Tutorial 6
   1. Use a Karnaugh map to simplify ABC  ABC  ABC
   2. In a certain chemical-processing plant, a liquid chemical is used in a manufacturing process.
      The chemical is stored in three different tanks. A level sensor in each tank produces a HIGH
      voltage when the level of chemical in the tank drops below a specified point. Design a circuit
      that monitors the chemical level in each tank and indicates when the level in any two tanks
      drops below the specified point. (hint: use Karnaugh Map to Simplify expression)
   3. Simplify the following circuit.




                                                                                                       b)




                                                                                                         c)
                      a)


   4. State the properties of TTL and CMOS logic gates.




  Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation –
                                                  G6601/S-I/Y-09